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All trigonometric numbers – sines or cosines of rational multiples of 360° – are algebraic numbers (solutions of polynomial equations with integer coefficients); moreover they may be expressed in terms of radicals of complex numbers; but not all of these are expressible in terms of real radicals. When they are, they are expressible more specifically in terms of square roots.

All values of the sines, cosines, and tangents of angles at 3° increments are expressible in terms of square roots, using identities – the half-angle identity, the double-angle identity, and the angle addition/subtraction identity – and using values for 0°, 30°, 36°, and 45°. For an angle of an integer number of degrees that is not a multiple 3° (π/60radians), the values of sine, cosine, and tangent cannot be expressed in terms of real radicals.

According to Niven's theorem, the only rational values of the sine function for which the argument is a rational number of degrees are 0, 1/2, 1, −1/2, and −1.

According to Baker's theorem, if the value of a sine, a cosine or a tangent is algebraic, then the angle is either a rational number of degrees or a transcendental number of degrees. That is, if the angle is an algebraic, but non-rational, number of degrees, the trigonometric functions all have transcendental values.

The list in this article is incomplete in several senses. First, the trigonometric functions of all angles that are integer multiples of those given can also be expressed in radicals, but some are omitted here.

Second, it is always possible to apply the half-angle formula to find an expression in radicals for a trigonometric function of one-half of any angle on the list, then half of that angle, etc.

Third, expressions in real radicals exist for a trigonometric function of a rational multiple of π if and only if the denominator of the fully reduced rational multiple is a power of 2 by itself or the product of a power of 2 with the product of distinct Fermat primes, of which the known ones are 3, 5, 17, 257, and 65537.

Fourth, this article only deals with trigonometric function values when the expression in radicals is in real radicals – roots of real numbers. Many other trigonometric function values are expressible in, for example, cube roots of complex numbers that cannot be rewritten in terms of roots of real numbers. For example, the trigonometric function values of any angle that is one-third of an angle θ considered in this article can be expressed in cube roots and square roots by using the cubic equation formula to solve

In the entries below, when a certain number of degrees is related to a regular polygon, the relation is that the number of degrees in each angle of the polygon is (n – 2) times the indicated number of degrees (where n is the number of sides). This is because the sum of the angles of any n-gon is 180° × (n – 2) and so the measure of each angle of any regular n-gon is 180° × (n – 2) ÷ n. Thus for example the entry "45°: square" means that, with n = 4, 180° ÷ n = 45°, and the number of degrees in each angle of a square is (n – 2) × 45° = 90°.

For cube roots of non-real numbers that appear in this table, one has to take the principal value, that is the cube root with the largest real part; this largest real part is always positive. Therefore, the sums of cube roots that appear in the table are all positive real numbers.

The derivation of sine, cosine, and tangent constants into radial forms is based upon the constructibility of right triangles.

Here right triangles made from symmetry sections of regular polygons are used to calculate fundamental trigonometric ratios. Each right triangle represents three points in a regular polygon: a vertex, an edge center containing that vertex, and the polygon center. An n-gon can be divided into 2n right triangles with angles of 180/n, 90 − 180/n, 90} degrees, for n in 3, 4, 5, …

Constructibility of 3, 4, 5, and 15-sided polygons are the basis, and angle bisectors allow multiples of two to also be derived.

Nonconstructible (with whole or half degree angles) – No finite radical expressions involving real numbers for these triangle edge ratios are possible, therefore its multiples of two are also not possible.

The induction above can be applied in the same way to all the remaining Fermat primes (F3=223+1=28+1=257 and F4=224+1=216+1=65537), the factors of π whose cos and sin radical expressions are known to exist but are very long to express here.

(Alternatively, without using Ptolemy's theorem, label as X the intersection of AC and BD, and note by considering angles that triangle AXB is isosceles, so AX = AB = a. Triangles AXD and CXB are similar, because AD is parallel to BC. So XC = a·(a/b). But AX + XC = AC, so a + a2/b = b. Solving this gives a/b = 1/φ, as above).

Alternately, the multiple-angle formulas for functions of 5x, where x ∈ {18, 36, 54, 72, 90} and 5x ∈ {90, 180, 270, 360, 450}, can be solved for the functions of x, since we know the function values of 5x. The multiple-angle formulas are:

If the denominator is a square root, multiply the numerator and denominator by that radical.

If the denominator is the sum or difference of two terms, multiply the numerator and denominator by the conjugate of the denominator. The conjugate is the identical, except the sign between the terms is changed.

If there is a complicated term, with only one kind of radical in a term, this plan may help. Square the term, combine like terms, and take the square root. This may leave a big radical with a smaller radical inside, but it is often better than the original.